HMMT 二月 2024 · 几何 · 第 3 题

HMMT February 2024 — Geometry — Problem 3

Let Ω and ω be circles with radii 123 and 61 , respectively, such that the center of Ω lies on ω . A chord of Ω is cut by ω into three segments, whose lengths are in the ratio 1 : 2 : 3 in that order. Given that this chord is not a diameter of Ω , compute the length of this chord.

解析

Let Ω and ω be circles with radii 123 and 61 , respectively, such that the center of Ω lies on ω . A chord of Ω is cut by ω into three segments, whose lengths are in the ratio 1 : 2 : 3 in that order. Given that this chord is not a diameter of Ω , compute the length of this chord. Proposed by: Benjamin Kang, Holden Mui, Pitchayut Saengrungkongka Answer: 42 Solution: Denote the center of Ω as O . Let the chord intersect the circles at W, X, Y, Z so that W X = t , XY = 2 t , and Y Z = 3 t . Notice that Y is the midpoint of W Z ; hence OY ⊥ W XY Z . ◦ The fact that ∠ OY X = 90 means X is the antipode of O on ω , so OX = 122 . Now applying power of point to X with respect to Ω gives 2 2 2 245 = 123 − OX = W X · XZ = 5 t = ⇒ t = 7 . Hence the answer is 6 t = 42 . Z Y O X W